An alpine rescue team is using a slingshot to send an emergency medical packet to climbers stranded on a ledge. (height=252 m, horizontal distance=416 m, landing angle= 75° below the horizontal).
and the question/
To calculate the initial speed with which the medical packet is launched using the slingshot, you can use the kinematic equations of motion. Here's how you can do it step by step:
Step 1: Determine the initial vertical velocity component (Vy):
Given that the packet is launched with a landing angle of 75° below the horizontal, we can calculate the vertical component of the initial velocity (Vy). We can use the horizontal distance (416 m) and the vertical height (252 m) as follows:
First, determine the time of flight (t) in seconds:
Using the equation: Δy = Vy * t + 0.5 * g * t^2
where Δy is the vertical displacement (252 m) and g is the acceleration due to gravity (approximately 9.8 m/s^2), we can solve for t.
252 = Vy * t + 0.5 * 9.8 * t^2
Now, rearrange the equation:
0.5 * 9.8 * t^2 + Vy * t - 252 = 0
This is a quadratic equation in t. Solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 0.5 * 9.8, b = Vy, and c = -252.
By substituting these values into the quadratic formula, we can find two values of t, one of which will be negative. Select the positive value of t.
Step 2: Calculate the initial horizontal velocity component (Vx):
Now that we have the time of flight (t), we can calculate the horizontal component of the initial velocity (Vx). We can use the equation:
Vx = (horizontal distance) / (time of flight)
Vx = 416 / t
Step 3: Calculate the initial velocity (V0):
Using the initial vertical velocity component (Vy) and the initial horizontal velocity component (Vx), we can calculate the initial velocity (V0) using the Pythagorean theorem:
V0^2 = Vx^2 + Vy^2
V0 = √(Vx^2 + Vy^2)
Substitute the values of Vx and Vy into the equation and calculate V0.
Step 4: Determine the initial launch angle (θ):
Given that the packet is launched at 75° below the horizontal, the angle of launch (θ) would be 180° - 75° = 105°.
So, the initial velocity (V0) and the launch angle (θ) are now known.
Note: It's important to note that in this scenario, the slingshot is not a typical projectile motion problem since the launch angle is below the horizontal. Projectile motion problems usually involve launching objects above the horizontal. However, the same principles of solving for initial velocity and launch angle still apply.
Now you have all the steps needed to calculate the initial speed (V0) with which the medical packet is launched using the slingshot.